y (354) 



344 ./, W. Gihhs — Equilihrhmi of Heterogeneous ISuhstemces. 



body. When the differential coefficients of the first order do not 

 vary sensibly except for distances greater than the radius of sensible 

 molecular action, we niaj^ regard them as completely determining the 

 state of strain of any element. There are nine of these differential 

 coefficients, viz., 



dx dx dx 



dx'' dy'"' cfe" 



dy dy dy 



d^" dy" ~dz" 



dz dz dz 



d^>' dy" W 



It will be observed that these quantities determine the orientation of 

 the element as well as its strain, and both these particulars must be 

 given in order to determine the nine differential coefficients. There- 

 fore, since the orientation is capable of three independent variations, 

 which do not affect the strain, the strain of the element, considered 

 without regard to directions in space, must be capable of six indepen- 

 dent variations. 



The physical state of any given element of a solid in any unvary- 

 ing state of strain is capable of one variation, which is produced by 

 addition or subtraction of heat. If we write fv» •'^•icl Vvi for the 

 energy and entropy of the element divided by its volume in the 

 state of reference, we shall have for any constant state of strain 



But if the strain varies, we may consider e^, as a function of //v, and 

 the nine quantities in (354), and may write 



dx dy dz 



where JY'x,, . . . Zy, denote the differential coefficients of £v» taken 



witli respect to -^, . . . ~^,. The physical signification of these 



quantities will be apparent, if we apply the formula to an element 

 which in the state of reference is a right parallelopiped having the 

 edges dx\ dy', dz\ and suppose that in the strained state the face in 

 which x' has the smaller constant value remains fixed, while the 

 opposite face is moved parallel to the axis of A'. If we also suppose 



